A026920 -id:A026920 - cp's OEIS frontend

A026923 Number of partitions of n into an odd number of parts, the greatest being 3; also, a(n+5) = number of partitions of n+2 into an even number of parts, each <= 3.

0, 0, 1, 0, 1, 1, 3, 2, 4, 3, 6, 5, 8, 7, 11, 9, 13, 12, 17, 15, 20, 18, 24, 22, 28, 26, 33, 30, 37, 35, 43, 40, 48, 45, 54, 51, 60, 57, 67, 63, 73, 70, 81, 77, 88, 84, 96, 92, 104, 100, 113, 108, 121, 117, 131

Offset: 1

Clark Kimberling

nonn

			Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
                                                          1+1+8
                                                   1+1+7  1+2+7
                                                   1+2+6  1+3+6
                                            1+1+6  1+3+5  1+4+5
                                     1+1+5  1+2+5  1+4+4  2+2+6
                              1+1+4  1+2+4  1+3+4  2+2+5  2+3+5
                       1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4
         1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...
-----------------------------------------------------------------------
  n  |     3      4      5      6      7      8      9     10      ...
-----------------------------------------------------------------------
a(n) |     1      0      1      1      3      2      4      3      ...
-----------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 06 2019

R. J. Mathar, Table of n, a(n) for n = 1..1000

3rd column of A026920.

Cf. A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309690, A309692, A309694.

A026923 := proc(n)
    local a,p1,p2,p3 ;
    a := 0 ;
    for p1 from 0 to n do
        for p2 from 0 to (n-p1)/2 do
            p3 := (n-p1-2*p2)/3 ;
            if type(p3,'integer') and p3 >=1 and type(p1+p2+p3,'odd') then
                a := a+1 ;
            end if:
        end do:
    end do:
    a;
end proc: # R. J. Mathar, Aug 22 2019

a(n) + A026927(n) = A069905(n). - R. J. Mathar, Aug 22 2019
Conjectures from Colin Barker, Sep 01 2019: (Start)
G.f.: x^3*(1 - x + x^2 + x^4) / ((1 - x)^3*(1 + x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) + a(n-6) - a(n-7) - a(n-10) + a(n-11) for n>11.
(End)

A026921 Triangular array E by rows: E(n,k) = number of partitions of n into an even number of parts, the greatest being k.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 0, 2, 1, 2, 1, 1, 0, 1, 2, 3, 2, 2, 1, 1, 0, 0, 2, 3, 3, 2, 2, 1, 1, 0, 1, 2, 5, 4, 4, 2, 2, 1, 1, 0, 0, 3, 4, 6, 4, 4, 2, 2, 1, 1, 0, 1, 3, 7, 7, 7, 5, 4, 2, 2, 1, 1, 0, 0, 3, 6, 10, 8, 7, 5, 4, 2, 2, 1, 1, 0, 1, 3, 9, 11, 13, 9, 8, 5, 4, 2, 2, 1, 1, 0

Offset: 1

Clark Kimberling

The reversed rows (see example) stabilize to A027193. [Joerg Arndt, May 12 2013]

			G.f. = (1)*q^0 +
(0) * q^1 +
(1 + 0*z) * q^2 +
(0 + 1*z + 0*z^2) * q^3 +
(1 + 1*z + 1*z^2 + 0*z^3) * q^4 +
(0 + 1*z + 1*z^2 + 1*z^3 + 0*z^4) * q^5 +
(1 + 1*z + 2*z^2 + 1*z^3 + 1*z^4 + 0*z^5) * q^6 +
(0 + 2*z + 1*z^2 + 2*z^3 + 1*z^4 + 1*z^5 + 0*z^6) * q^7 +
... [_Joerg Arndt_, May 12 2013]
Triangle starts:
01: [0]
02: [1, 0]
03: [0, 1, 0]
04: [1, 1, 1, 0]
05: [0, 1, 1, 1, 0]
06: [1, 1, 2, 1, 1, 0]
07: [0, 2, 1, 2, 1, 1, 0]
08: [1, 2, 3, 2, 2, 1, 1, 0]
09: [0, 2, 3, 3, 2, 2, 1, 1, 0]
10: [1, 2, 5, 4, 4, 2, 2, 1, 1, 0]
11: [0, 3, 4, 6, 4, 4, 2, 2, 1, 1, 0]
12: [1, 3, 7, 7, 7, 5, 4, 2, 2, 1, 1, 0]
13: [0, 3, 6, 10, 8, 7, 5, 4, 2, 2, 1, 1, 0]
14: [1, 3, 9, 11, 13, 9, 8, 5, 4, 2, 2, 1, 1, 0]
15: [0, 4, 8, 14, 14, 14, 9, 8, 5, 4, 2, 2, 1, 1, 0]
16: [1, 4, 12, 16, 20, 17, 15, 10, 8, 5, 4, 2, 2, 1, 1, 0]
17: [0, 4, 11, 20, 22, 23, 18, 15, 10, 8, 5, 4, 2, 2, 1, 1, 0]
18: [1, 4, 15, 23, 30, 28, 26, 19, 16, 10, 8, 5, 4, 2, 2, 1, 1, 0]
19: [0, 5, 13, 28, 33, 37, 31, 27, 19, 16, 10, 8, 5, 4, 2, 2, 1, 1, 0]
20: [1, 5, 18, 31, 44, 44, 43, 34, 28, 20, 16, 10, 8, 5, 4, 2, 2, 1, 1, 0]
... [_Joerg Arndt_, May 12 2013]

E(n, k) = O(n-k, 1)+O(n-k, 2)+...+O(n-k, m), where m=MIN{k, n-k}, n >= 2, O given by A026920.

Columns k=3..6: A026927, A026928, A026929, A026930.

N = 20;  q = 'q + O('q^N);
gf = sum(n=0,N, q^(2*n)/prod(k=1, 2*n, 1-'z*q^k) );
v = Vec(gf);
{ for(n=2, #v, /* print triangle starting with row 1: */
    p = Pol('c0 +'cn*'z^n + v[n],'z);
    p = polrecip(p);
    p = Vec(p);
    p[1] -= 'c0;
    p = vector(#p-2, j, p[j]);
    print(p);
); }
/* Joerg Arndt, May 12 2013 */

G.f. (including term a(0)=1): sum(n>=0, q^(2*n)/prod(k=1..2*n, 1-z*q^k) ), set z=1 to obtain g.f. for A027187. [Joerg Arndt, May 12 2013]

A026920(n,k) + E(n,k) = A008284(n,k). - R. J. Mathar, Aug 23 2019

A026922 Number of partitions of n into an odd number of parts, the greatest being 2; also, a(n+3) = number of partitions of n+1 into an even number of parts, each <=2.

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 7, 8, 8, 9, 8, 9, 9, 10, 9, 10, 10, 11, 10, 11, 11, 12, 11, 12, 12, 13, 12, 13, 13, 14, 13, 14, 14, 15, 14, 15, 15, 16, 15, 16, 16, 17, 16, 17, 17, 18, 17, 18, 18, 19, 18, 19, 19, 20, 19, 20, 20, 21

Offset: 1

Clark Kimberling

a(n) is also the number of partitions of n into two parts, the larger being odd (the conjugate of the defining partition). Example: a(10) = 3 because we have 55, 73 and 91. - Emeric Deutsch, Nov 12 2008

			a(10)=3 because we have 22222, 2221111 and 211111111. - _Emeric Deutsch_, Nov 12 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Column 2 of A026920. Essentially the same as A008624.

G:=x^2*(x^2-x+1)/((x+1)*(1-x)^2*(x^2+1)): Gser:= series(G,x=0,105): seq(coeff(Gser,x,n), n=1..82); # Emeric Deutsch, Nov 12 2008
a := proc(n): if (n mod 4 = 3) then floor((n+2)/4) - 1 else floor((n+2)/4) fi: end: seq(a(n), n=1..82); # Johannes W. Meijer, Oct 10 2013

CoefficientList[Series[x (1 - x + x^2) / (1 - x - x^4 + x^5), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 15 2013 *)

{a(n) = n \ 2 - ((n + 1) \ 4)} /* Michael Somos, Oct 14 2008 */

a(2*n + 1) = a(2*n - 2) = A004526(n).
a(n) = floor((n+2)/4) - [n == 3 mod 4] = floor((1/8)*{2*n - 1 + 3*(-1)^n + 2*(-1)^[(n-1)/2]}). - Ralf Stephan, Jun 09 2005
a(n) = A008624(n-2).  - R. J. Mathar, Oct 23 2008
From Emeric Deutsch, Nov 12 2008: (Start)
G.f. = sum(sum(x^(2*i-1+j), j=1..2*i-1), i=1..infinity).
G.f. = x^2*(1-x+x^2)/[(1+x)*(1-x)^2*(1+x^2)]. (End)
From Michael Somos, Oct 14 2008: (Start)
Euler transform of length 6 sequence [ 0, 1, 1, 1, 0, -1].
a(n) = a(n-1) + a(n-4) - a(n-5). a(1 - n) = -a(n).
G.f.: x^2 * (1 - x + x^2) / (1 - x - x^4 + x^5). (End)

More terms from Emeric Deutsch, Nov 12 2008

A026924 Number of partitions of n into an odd number of parts, the greatest being 4; also, a(n+7) = number of partitions of n+3 into an even number of parts, each <=4.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 3, 3, 5, 5, 8, 8, 12, 13, 18, 19, 24, 26, 33, 35, 43, 46, 55, 59, 69, 74, 86, 91, 104, 111, 126, 134, 150, 159, 177, 187, 207, 219, 241, 254, 277, 292, 318, 334, 362, 380, 410, 430, 462, 484, 519, 542, 579, 605

Offset: 1

Clark Kimberling

nonn

R. J. Mathar, Table of n, a(n) for n = 1..394

4th column of A026920.

A026924 := proc(n)
    local a,p1,p2,p3,p4 ;
    a := 0 ;
    for p1 from 0 to n do
        for p2 from 0 to (n-p1)/2 do
            for p3 from op(1+modp(n-p1-2*p2,4),[0,3,2,1]) to (n-p1-2*p2)/3 by 4 do
                p4 := (n-p1-2*p2-3*p3)/4 ;
                if type(p4,'integer') and p4 >=1 and type(p1+p2+p3+p4,'odd') then
                    a := a+1 ;
                end if:
            end do:
        end do:
    end do:
    a;
end proc: # R. J. Mathar, Aug 22 2019

a(n) + A026928(n) = A026810(n). - R. J. Mathar, Aug 22 2019

A026926 Number of partitions of n into an odd number of parts, the greatest being 6; also, a(n+11) = number of partitions of n+5 into an even number of parts, each <=6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 3, 6, 7, 11, 12, 18, 21, 30, 34, 46, 54, 70, 80, 101, 116, 143, 163, 197, 225, 269, 303, 357, 403, 469, 525, 606, 677, 774, 860, 976, 1082, 1221, 1346, 1509, 1661, 1852, 2029, 2252, 2462, 2720, 2964, 3261

Offset: 1

Clark Kimberling

nonn

6th column of A026920.

Table[Length[Select[IntegerPartitions[n],#[[1]]==6&&OddQ[Length[#]]&]],{n,60}] (* Harvey P. Dale, Dec 31 2022 *)

A026931 Triangular array T read by rows: T(n,k) = number of partitions of n into an odd number of parts, each <= k.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 3, 3, 4, 0, 2, 3, 4, 4, 5, 1, 2, 5, 6, 7, 7, 8, 0, 2, 4, 7, 8, 9, 9, 10, 1, 3, 7, 10, 13, 14, 15, 15, 16, 0, 3, 6, 11, 14, 17, 18, 19, 19, 20, 1, 3, 9, 14, 20, 23, 26, 27, 28, 28, 29, 0, 3, 8, 16, 22, 28, 31, 34, 35, 36

Offset: 1

Clark Kimberling

T(n, k) = Sum_{i=1..k} A026920(n, i).

cp's OEIS Frontend

A026923 Number of partitions of n into an odd number of parts, the greatest being 3; also, a(n+5) = number of partitions of n+2 into an even number of parts, each <= 3.

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A026921 Triangular array E by rows: E(n,k) = number of partitions of n into an even number of parts, the greatest being k.

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A026922 Number of partitions of n into an odd number of parts, the greatest being 2; also, a(n+3) = number of partitions of n+1 into an even number of parts, each <=2.

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A026924 Number of partitions of n into an odd number of parts, the greatest being 4; also, a(n+7) = number of partitions of n+3 into an even number of parts, each <=4.

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A026926 Number of partitions of n into an odd number of parts, the greatest being 6; also, a(n+11) = number of partitions of n+5 into an even number of parts, each <=6.

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Crossrefs

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Mathematica

A026931 Triangular array T read by rows: T(n,k) = number of partitions of n into an odd number of parts, each <= k.

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